Problem 164 - Precalculus

Question

The line \(y=\frac{-3}{7} x\) passes through the origin in the \(x, y\) -plane. What is the measure of the angle that the line makes with the negative \(x\) -axis?

Step-by-Step Solution

Answer

The angle is approximately \(156.8^\circ\).

Step 1 - Understand the slope

The slope of the line is given as \( -\frac{3}{7} \). This slope value determines the angle that the line makes with the positive \(x\)-axis.

Step 2 - Find the angle with the positive x-axis

To find the angle \( \theta \) that the line makes with the positive \(x\)-axis, use the formula \( \tan(\theta) = \text{slope} = -\frac{3}{7} \). We calculate \( \theta = \arctan(-\frac{3}{7}) \) using a calculator to find the angle in degrees.

Step 3 - Calculate the angle

Using a calculator, \( \arctan(-\frac{3}{7}) \) gives \( \theta \approx -23.2^\circ \). This is the angle with the positive \(x\)-axis.

Step 4 - Find the angle with the negative x-axis

To find the angle with the negative \(x\)-axis, we need to add \(180^\circ\) to the positive \(x\)-axis angle. So, the angle with the negative \(x\)-axis is \(-23.2^\circ + 180^\circ = 156.8^\circ\).

Key Concepts

Angle with X-axis

In trigonometry, the angle a line makes with the x-axis is a crucial concept. When examining a line on the coordinate plane, it is useful to measure this angle because it helps to understand the line's direction and orientation.
The x-axis is typically considered the horizontal line, and angles are measured from this axis. For a line given by an equation, such as the one in the exercise, you often start by finding out how steeply the line angles away from the x-axis.
Angles that are measured counterclockwise from the positive x-axis are considered positive, whereas angles measured clockwise are negative. By standard, the angle with the positive x-axis is measured first, and adjustments are made to determine angles in other orientations, like with the negative x-axis.

Slope of a Line

The slope of a line is a measure of how steep a line is. It indicates the rise and run between points on the line. In simple terms, it is how much the y-value of a point on the line changes for a change in the x-value.
For the line in our exercise, the slope is \(-\frac{3}{7}\). A negative slope like this means that as you move from left to right along the line, the line goes downward.
The slope can be thought of in terms of a ratio: the numerator (here, -3) indicates a fall, and the denominator (7) indicates the run. This slope also ties directly into calculating angles using trigonometric functions, as it is used in our next concept, the arctan function, to find angles with respect to the x-axis.

Arctan Function

The arctan function, also known as inverse tangent, is a fundamental tool in trigonometry used to find the angle from the slope or tangent of a trigonometric identity. This function helps in finding the angle made by a line with the x-axis by using its slope.
To find an angle \(\theta\) with a known slope \(m\), you use the formula \(\theta = \arctan(m)\). In our exercise, we find the angle \(\theta\) made by the line with the positive x-axis by calculating \(\arctan(-\frac{3}{7})\), which approximately results in \(-23.2^\circ\).
Arctan will return angles between \(-90^\circ\) and \(90^\circ\), which corresponds to the possible values of a slope in this range. Remember, to get an angle with respect to the negative x-axis, you can adjust by adding \(180^\circ\). This adjustment provides you with the full, extended direction of the line.